Advances in Possible Orders of Circulant Hadamard Matrices And

Advances in Possible Orders of Circulant Hadamard Matrices, and Sequences with Large Merit Factor

Jason Hu1 Brooke Logan2

1Department of Mathematics University of California, Berkeley

2Department of Mathematics Rowan University

August 7, 2014 Outline

Introduction Circulant Hadamard Matrices The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results New Family of Binary Sequences Note: A generalization of Galois sequences New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Autocorrelations

Deﬁnition (Aperiodic Autocorrelation) of a sequence of length n at shift k, 0 ≤ k < n is

n−1−k X ck = ai a¯i+k i=0

Deﬁnition (Periodic Autocorrelation) of a sequence of length n at shift k, 0 ≤ k < n is

n−1 X γk = ai a¯i+kmod(n) i=0 Barker Sequences

Deﬁnition A barker sequence is a binary sequence {a0, a1, ...an−1} of length n such that when calculating the sequence’s aperiodic autocorrelation at shift k = 0, c0 = n and for shift ranging from 1 ≤ k < n the aperiodic autocorrelation is |ck | ≤ 1 Example ({1, 1, −1})

3−1−0 X c0 = ai ai+0 = 1(1) + 1(1) + (−1)(−1) = 3 i=0 3−1−1 X c1 = ai ai+1 = 1(1) + 1(−1) = 0 i=0 3−1−2 X c2 = ai ai+2 = 1(−1) = −1 i=0 Barker Sequence! Barker Conjecture

There exists no Barker sequence of n > 13

Proven for

n of odd length 33 even n = 3979201339721749133016171583224100 or n > 4 ∗ 10 (P. Borwein & M. Mossinghoﬀ, 2014) Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Circulant Hadamard Matrices

Deﬁnition (Hadamard Matrix) T An n × n matrix H of ±1 where HH = nIn

Deﬁnition (Circulant Matrix) A matrix where each row after the ﬁrst row is one cyclic shift to the right of the previous row. Example (Circulant Hadamard Matrix)

+++- − + + + + − + + + + − + Circulant Hadamard Matrices

Deﬁnition (Hadamard Matrix) T An n × n matrix H of ±1 where HH = nIn

Deﬁnition (Circulant Matrix) A matrix where each row after the ﬁrst row is one cyclic shift to the right of the previous row. Example (Circulant Hadamard Matrix)

+++-  + − + + 4 0 0 0 − + + + + + − + 0 4 0 0   ·   =   + − + + + + + − 0 0 4 0 + + − + - + + + 0 0 0 4 Relationship between Barker sequences and Circulant Hadamard matrices

Deﬁnition (Circulant Hadamard Conjecture) There exists no Circulant Hadamard Matrix with n > 4

Barker Sequence ⇒ small aperiodic autocorrelations ⇒ small periodic autocorrelations ⇒ Circulant Hadamard Matrix Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Restrictions

By Deﬁnition of a Hadamard Matrix

Must be a multiple of 4 (or n = 1, 2)

Turyn, 1965

assuming n>2 then 2 n = 4m m is odd m cannot be a prime power more to come n = 4m2

When searching in a given bound M:

m = p1p2...pu ≤ M (1)

Theorem (Turyn) 2 1 p ≤ (2M ) 3 n = 4M2

When searching in a given bound M:

p1 → p2 → ... → p1 Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Links

Ascending: q→p

q < p and qp−1 ≡ 1 mod p2

Descending: p→q

q < p and pq−1 ≡ 1 mod q2

Flimsy: p q q|(p − 1) Types of Ascending Pairs

Diﬀerent Cases q p Worst Case Scenario 13 Previous Search M = 10 1 M 2 3 q < p ≤ min( q , (2M ) ) M = 5 ∗ 1014

Double Wieferich Prime Pair q p

M 2 1 q < p ≤ min( , (2M ) 3 ) q

Theorem (W. Keller & J. Richstein ) q 2 Let p1 be a primitive root of the prime q and define p2 = p1 mod q . m 2 Then {p2 mod q : m = 0, 1, .., q − 2} represents a complete set of incongruent solutions of pq−1 ≡ 1(mod q2), each of which generates an infinite sequence of solutions in arithmetic progression with difference q2

pq−1 ≡ 1 mod q2 Example

q = 83

p1 = PrimitiveRoot(q) = 2

Primitive root generator of the multiplicative group mod p q 2 p2 = p1 mod q = 1081 m 2 q−3 p2 mod q , m = 0, 1, .., 2 m 2 2 Even case: a = (p2) + q and b = q − a m 2 Odd Case: a = (p2) and b = 2q − a 2 a, a + 2q , ... ,

m = 37 ⇒ a = 4871 Ascending and Flimsy q → p and p q

M 2 1 q < p ≤ min( , (2M ) 3 ) 3q q|(p − 1) qp−1 ≡ 1(mod p2) Special Ascending

Double Wieferich Prime Pairs Ascending and Flimsy 3 1006003 3 → 1006003 5 1645333507 5 → 20771 83 4871 5 → 53471161 911 318917 13 → 1747591 2903 18787 44963 → 5395561 Strictly Ascending

(3→11→71→3)→...→(q→p) r → q → p → r Strictly Ascending

M q < p ≤ 3∗11∗71∗q

r → q → p → r Strictly Ascending

M q < p ≤ 3∗11∗71∗q

M q < p ≤ r∗q

M q < p ≤ q2 Strictly Ascending

1 M M M 2 3 q < p ≤ min(max(3∗11∗71∗q , r∗q , q2 ), (2M ) ) Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons 1: List A and List B = All primes in Ascending Pairs 2: while Length B > 0 do 3: for p ∈ B do 4: for All Primes, q, such that 3 ≤ q < p do 5: if pq−1 ≡ 1 mod q2 then 6: Add (p, q) to solid link list and add q to T 7: else if q|(p − 1) then 8: Add (p, q) to ﬂimsy link list and add q to T 9: end if 10: end for 11: end for 12: B = T /A, A = A ∪ B, and Clear T 13: end while Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Comparing results

M Bound 1013 5 ∗ 1014 Vertices 643931 15342 Ascending 59837 6616 Descending 1673025 33935 Flimsy 1729116 33264 Creating Circuits

Johnson’s Circuit Finding Algorithm and Augmenter= 501630

F Test 13 M = 10 cycles 2064 14 M = 5 ∗ 10 cycles 6683

Turyn Test

Leung Schmidt Test Theorem 1,5,10 F-Test Leung and Schmidt 2005

vp(m) = multiplicity of p in factorization of m Q mq = q-free and squarefree part of m: mq = p|m,p6=q p p−1 b(p, m) = maxq|m,q≤p{vp(q − 1) + vp(ordmq (q))} 2 Q b(p,m) F (m) = gcd(m , p|m p )

Theorem If n = 4m2 is the order of a circulant Hadamard matrix, then F (m) ≥ mφ(m) Turyn Test

Deﬁnition a is semi-primitive mod b: aj ≡ −1 mod b for some j Deﬁnition r is self-conjugate mod s: For each p|r, p is semi-primitive mod the p-free part of s. Theorem If n = 4m2 is the order of a Circulant Hadamard Matrix, r|m, s|n, gcd(r, s) has k ≥ 1 distinct prime divisors, and r is self-conjugate mod s, then rs ≤ 2k−1n Turyn Test

Theorem If n = 4m2 is the order of a Circulant Hadamard Matrix, r|m, s|n, gcd(r, s) has k ≥ 1 distinct prime divisors, and r is self-conjugate mod s, then rs ≤ 2k−1n

Lm = {p1, p2, ..., pu}

Ln = {2, 2, p1, p2, ..., pu, p1, p2, ..., pu}

Let α ∈ Lm and β ∈ Ln Take α ∩ β If rs > 2k−1n and r self-conjugate mod s ->throw it out Cycles that Fail

Length Starting Value Turyn LS5 LS10 LS1 Surviving 2 5 5 - - - 0 3 59 50 0 0 0 9 4 296 192 5 1 9 87 5 915 6 1744 7 1946 8 1229 9 430 10 59 1st New Cycle! m=10010975913705 Largest m value cycle found m=499317956344211 Barker

Given (p, q), both p and q must be ≡ 1 mod 4 Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Merit Factor d−1 Given a sequence A = {ak }k=0 of length d, recall that the aperiodic autocorrelation at shift u is

n−d−1 X cu = aj aj+u j=0

Deﬁnition The merit factor of A is deﬁned to be d2 MF (A) = Pd−1 2 2 u=0 |cu|

Engineering Application: measures how large peak energy of a signal is compared to total sidelobe energy Binary vs Polyphase

Binary and polyphase sequences with large merit factor are useful!

Binary sequence

Values in {+1, −1} No known inﬁnite family of sequences for which merit factor grows without bound

Polyphase sequence 2πi q Values in {e N | q ∈ Z}, for some ﬁxed N Known families of sequences with unbounded (polynomial) merit factor growth Motivation: Barker Sequences - Binary Sequences with Best Merit Factor?

Barker sequences appear to have the largest merit factors relative to the length of the sequence. MF 14

Length 2 4 6 8 10 12

Sadly, no Barker sequence of length N > 13 is known to exist. Merit Factor Problem

Let An be the set of all binary sequences of length n. Deﬁnition

Fn := max MF (A), A∈An the maximal value of the merit factor among sequences of length n.

Open Question (The Merit Factor Problem)

What is lim supn→∞ Fn? Families of Binary sequences with Large Merit Factor

Asymptotically, merit factor of these sequences is a relatively large constant:

Legendre: 3 Galois: 3 Rudin-Shapiro: 3

Rotated Legendre: 6 Rotated and truncated Legendre: 6.34

Such inﬁnite families are relatively hard to ﬁnd Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons A generalization of Galois sequences

Galois sequences - binary sequences based on canonical additive characters of Galois extensions of F2

We extend to other prime bases (yielding polyphase sequences) and observe similar behavior

Deﬁnition For prime p and m ≥ 2, consider the Galois extension Fpm over Fp. The relative trace of Fpm over Fp is

Tr: Fpm −→ Fp m−1 X j β 7−→ βp j=0 2πi ∗ Let ζ = e p , and θ be a primitive element of the group (Fpm ) . Deﬁnition The canonical additive character of Fpm is given by

χ: Fpm −→ Fp c 7−→ ζTrc

Then the generalized Galois sequence of length pm with respect to θ is given by the coeﬃcient sequence of the polynomial

pm−2 X i i Yp,m,θ(z) = χ θ z i=0 Conjecture m Let yp,m,θ denote the generalized Galois sequence of length p with respect to θ. We conjecture that for a ﬁxed prime p, we have

lim MF (yp,m,θ) = 3 m→∞ and for a ﬁxed exponent m, we have

lim MF (yp,m,θ) = 3 p→∞ A New Family

based on rows of certain matrices

Deﬁnition A Walsh matrix is a square matrix of dimension 2k for some integer k ≥ 1, deﬁned recursively via

 1 1 1 1  1 1  1 −1 1 −1  H1 = H2 =   1 −1  1 1 −1 −1  1 −1 −1 1

Hk−1 Hk−1 Hk = = H1 ⊗ Hk−1 Hk−1 −Hk−1 Dimension 21

1 2

1 1

2 2

1 2 Dimension 22

1 2 3 4

1 1

2 2

3 3

4 4

1 2 3 4 Dimension 24

1 5 10 16

1 1

5 5

10 10

16 16

1 5 10 16 Dimension 28

1 100 200 256 1 1

100 100

200 200

256 256 1 100 200 256 Dimension 210

1 500 1024 1 1

500 500

1024 1024 1 500 1024 Above ordering for Walsh Matrix is natural

Corresponds to the Hadamard Transform (without normalization), equivalent to a multidimensional discrete Fourier transform (DFT) of size 2 × 2 × · · · × 2 | {z } n times

We use different ordering due to Bespalov, 2009 Definition A Walsh matrix in Bespalov ordering is a square matrix of dimension 2k for some integer k ≥ 1, defined recursively via

 1 1 1 1  1 1  1 −1 −1 1  V1 = V2 =   1 −1  1 −1 1 −1  1 1 −1 −1

Vn−1 Vn−1Pn−1 Vn = Pn−1Vn−1 −Pn−1Vn−1Pn−1, n n where Pn is a 2 × 2 {0, 1}-matrix with 1 on the secondary diagonal.

New family of binary sequences constructed by concatenating rows of Walsh matrices in Bespalov ordering Dimension 21

1 2

1 1

2 2

1 2 Dimension 22

1 2 3 4

1 1

2 2

3 3

4 4

1 2 3 4 Dimension 24

1 5 10 16

1 1

5 5

10 10

16 16

1 5 10 16 Dimension 28

1 100 200 256

1 1

100 100

200 200

256 256

1 100 200 256 Dimension 210

1 500 1024

1 1

500 500

1024 1024

1 500 1024 Dimension 212

1 1000 2000 3000 4096

1 1

1000 1000

2000 2000

3000 3000

4096 4096

1 1000 2000 3000 4096 Experimental Results

Merit factors of Walsh sequences in Bespalov enumeration, length 22n n MF 1 4.00000 2 3.20000 3 3.04762 4 3.01176 5 3.00293 6 3.00073 7 3.00018 8 3.00005 Experimental Results

6 Simulated annealing (10 trials) on the matrix row order did not ﬁnd any better rearrangements

Conjecture

Let {Bn} denote the family of sequences deﬁned via Bespalov’s enumeration of Walsh matrices. Then

lim MF (Bn) = 3 n→∞ Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons L4 norms of polynomials

Provides other viewpoint for Merit Factor Problem n−1 For each binary (resp. polyphase) sequence {ak }k=0, can form polynomials with binary (resp. unimodular) coeﬃcients

n−1 X k fn = ak z k=0

Deﬁnition p Let p ≥ 1. For a polynomial f ∈ C [z], its L norm on the complex unit circle is given by

1/p 1 Z 2π p iθ kf kp = f (e ) dθ 2π 0 L4 norm vs. Merit Factor

Let f ∈ C [z] be of degree d − 1 with coeﬃcient sequence A. By Parseval’s Theorem, write

2 kf k2 = d

Since z = 1/z on the unit circle,

d−1 d−1 4 2 X u 2 X 2 kf k4 = kf (z)f (z)k2 = cuz = d + 2 |cu| u=1−d u=0

Thus d2 kf k4 MF (A) = = 2 Pd−1 2 kf k4 − kf k4 2 u=0 |cu| 4 2 A question due to Littlewood

Question 4 4 How slowly can kpnk4 − kpnk2 grow for a sequence of polynomials {pn} with unimodular coeﬃcients and increasing degree? Theorem (Schmidt, 2013)

For the sequence {hN } of polynomials given by

N−1 N−1 X X jk jN+k hN (z) = ζ z , j=0 k=0 where ζ = e2πi/N , we have

4 4 khN k4 − khN k2 4 lim 3 = 2 . N→∞ khN k2 π A New Family of Polynomials

Deﬁnition 2πi πi For each integer N ≥ 1, write ζ = e N , ω = e N . Deﬁne a new family 2 of polynomials {fN }, where fN is of degree N − 1, given by

N−1 N−1 X X jN+k fN (z) = x(jN+k)z , j=0 k=0 where ( ζjk (−ω)j+k , if N is even xjN+k = ζjk (−ω)j (−1)k , if N is odd Same asymptotic behavior as that of {hN }: Proposition

4 4 kfN k4 − kfN k2 4 lim 3 = 2 N→∞ kfN k2 π Outline of Proposition

Use Schmidt’s method for determining the asymptotic behavior of the L4 norm:

4 Specializing our previous formula for the L norm,

N−1 N−1 4 4 X X 2 kfN k4 = N + 2 |cuN+v | u=0 v=0 Outline of proposition

(Long) algebraic manipulations on the second sum to obtain Lemma

 sin2 (2k−1)π  X X 2N N4 − N2 + 8N if N is even  sin2 πv  N 1≤k≤v N 4 1≤v≤ 2 kfN k4 = sin2 (2k−1)π  X X 2N N4 + 8N if N is odd  sin2 πv  N−1 1≤k≤v N 1≤v≤ 2 Use an analytic bound to conclude Lemma For N ≥ 1, we have

2 2k−1 X X sin π 4 8N 2N = N3 + O(N2) sin2 πv π2 N 1≤k≤v N 1≤v≤ 2

Thus

4 4 4 4 kfN k4 − kfN k2 kfN k4 − N 4 lim 3 = 3 = 2 . N→∞ kfN k2 N π Open Question Does there exist a sequence of polynomials with unimodular 4 coeﬃcients whose normalized asymptotic L4 norm is less than ? π2 Introduction Circulant Hadamard Matrices

The Search for Circulant Hadamard Matrices Restrictions on n Ascending Wieferich Prime Pairs Descending Wieferich Prime Pairs Comparing results

New Family of Binary Sequences Note: A generalization of Galois sequences

New Family of Polyphase Sequences L4 norms of polynomials Merit Factor Comparisons Some Other Polyphase Sequences with Large Merit Factor Growth Rate

2 A few length N sequences {xjN+k }0≤j,k

xjN+k = exp(πiφj,k ) where for

P1 Sequences: φj,k = −(N − 2j − 1)(jN + k)/N

Corrected Px Sequences: ( [(N − 1)/2 − k][N − 2j − 1] /N if N is even φj,k = [(N − 2)/2 − j][N − 2k − 1] /N if N is odd

Frank sequences: φj,k = 2jk/N (coeﬀ. sequences of Schmidt’s {hN } above) Comparison with other polyphase sequences

Length 5 10 15 20

Figure : Merit Factor of Sequences vs. Square Root of Length

Blue: New, Corrected Px Orange: Frank, P1 Red: P3, P4, Golomb, Chu Experiments on Lower Order Terms

Numerical calculations indicate that asymptotic behaviors of new/Px sequences and Frank sequences agree for order N2. 1.1 Conjectured that diﬀerence is at order about N Goals and Future Work

Binary merit factor conjectures: Walsh sequences in Bespalov ordering and generalized Galois sequences

Finding other binary sequences with large asymptotic merit factor

Does there exist an increasing sequence of polynomials with unimodular coeﬃcients whose normalized asymptotic L4 norm is 4 less than ? π2

The Merit Factor Problem: does the maximal value of the merit factor among sequences of length n have a limit as n grows without bound? Any Questions? Acknowledgement

We would like to thank

Professor Michael Mossinghoﬀ

ICERM and the NSF

Brown University Center for Computation and Visualization

TAs

Helpful comments from Dat Nguyen and Paxton Turner Bibliography (1)

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